Much of students' mathematical learning involves expanding understanding of a mathematical idea or relationship by shifting from one type of representation to a different representation of the same relationship. This is one of the reasons that it is important for students to use a variety of manipulative materials, which are then carefully related to paper-and-pencil methods of solving problems. Through this work, they move from informal representations to the more formal, and abstract representations that more advanced work will require.

A third-grade teacher introduced a unit of study on two-digit multiplication with an open-ended assignment for student pairs. The students were asked to think of a story problem where someone would want to know the product of 15 x 12 and to then show a method for finding that product:

Make up a story problem that would go with this number sentence: 15 x 12 = ?

Write your story problem.

Show a method for finding the product of 15 x 12.

Discuss your method with another group. What does your group's representation show differently? What is most clear?

Students suggested several different contexts and manipulative materials to be used. For example, one suggestion was 15 plastic bags with 12 crayons each (represented by 15 rectangles with "12 crayons" written on each rectangle). Another suggestion was a dot array of 15 teams lined up on the playground, with 12 players on each team. Another group of students worked first on their solution method and struggled to match 15 x 12 to a story problem. They used a method that had been used to introduce multiplication in the prior grade: making 15 towers of linking cubes with 12 cubes each and then finding an efficient method for counting them all:

The methods used for finding the product were somewhat dependent on the representation that was used. The group with the crayons added 12 plus 12 plus 12, etc. -- first mentally, then on paper -- to find the total; they had some difficulty keeping track of the number of times that 12 was added:

The two groups with the array of team members and with the towers of linking cubes both experimented with a variety of ways of finding the product before partitioning their rows of dots or towers of cubes into two parts, 10 and 2. They worked first with the rows or towers of 10, finding that partial product easily, and then found the partial product of 15 times 2.

As students invent their own way to show a relationship, such as 15 x 12, with materials, pictures, or diagrams, they engage in thought that helps strengthen their understanding of the operation of multiplication. When a number of different representations for a given problem are shared and discussed, similarities in the mathematical structure of each representation can be highlighted. For example, a student with 15 towers with 12 linking cubes in each tower can point out the similarity to a representation that uses 15 plastic bags with 12 crayons in each. Similarly, an area diagram can be related to an array that is made of 12 rows with 15 objects per row:

Notice that while an array representation may initially encourage students to use simple counting to solve the problem, an area model clearly shows the advantage of making smaller, easier to calculate groups or areas. This also very clearly corresponds to the standard algorithm for multiplication, as shown above.

During the following week, the teacher extended the example of the lines of team players and connected it to using base-ten blocks to represent 1, 10, or 100 players instead of drawing individual dots. Over the course of the next several weeks, through class discussion and guidance from the teacher, the class developed connections between this manipulative model, arrays drawn on grid paper, and symbolic methods for finding the product of two two-digit numbers. They also practiced mental-math methods for finding products by breaking a problem into two parts, such as (15 x 10) + (15 x 2).

"[D]ifferent representations support different ways of thinking about and manipulating mathematical objects. An object can be better understood when viewed through multiple lenses." (NCTM, 2000, p. 360)

Watch the video segment (duration 0:27) in the viewer box on the upper left to hear a reflection from Pam Hardaway, a middle school teacher in California. Her ideas about manipulative materials are applicable in grades 3-5 as well.